By L. Bieberbach
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Additional info for Conformal Mapping
W = zn) at w = 0 and w = 00. Let us now ask, what about the Preservation-of-Neighborhoods Theorem in connection with our function? It obviously applies without any modification to the neighborhood of any point other than zero and infinity. Peculiarities are encountered, however, when we deal with the mapping of neighborhoods of z = 0 and z = 00, or with the mapping of the whole z-plane. To be sure, the Riemann surface is a closed, connected point set; but it is not a region in the sense in which this term was defined in § 1.
For the proof, we refer the reader to more detailed exposiFIG. 8 tions (such as Vol. II of the author's Lehrbuch der Funktio1tentheorie, Chelsea Pub!. , New York 1945). Here we merely add that all the mappings of the group can be generated by composition from two of them, namely from the parabolic mapping z' = z + 1 and the elliptic mapping w = -l/z. The former maps one of the two boundary lines of the strip onto the other, while the latter has i and - i as its fixed points and maps the two arcs of the unit ~ .
We shall § 4. LINEAR FUNCTIONS 25 use 8 to stand for any linear function, whose inverse-that we have just seen how to calculate above-we shall then denote by 8-I, as is usual in algebra. The following further result is now almost immediate: THEOREM III: The composition of any number of non-constant linear functions always leads to further non-constant linear functions. In proof, let 8 1 === II (z) and 8 2 = l2 (z) ; then 8 1 8 2 stands for l1 (l2 (z) ). The inverse of this last is 8 21 8 t 1 • The determinant of 8 1 8 2 is the product of the determinants of 8 1 and 8 2 , and can not vanish since neither of the factors vanishes.